Strong diffusion approximation in averaging with dynamical systems fast motions

The paper deals with the fast-slow motions setups in the continuous time dXε(t)dt=1εΣ(Xε(t))ξ(t/ε2)+b(Xε(t),ξ(t/ε2)),t∈[0,T] and the discrete time X^ε((n + 1)ε²) = X^ε(nε²) + εΣ(X^ε(nε²))ξ(n) + ε²b(X^ε(nε²), ξ(n)), n = 0, 1,…, [T/ε²], where Σ and b are smooth matrix and vector functions and ξ is a stationary vector stochastic process with weakly dependent terms and such that Eξ(0) = 0. In fact, our conditions will enable us to reduce the setup to the case when Σ is the identity matrix. The assumptions imposed on the process ξ allow applications to a wide class of observables g in the dynamical systems setup so that ξ can be taken in the form ξ(t)= g(F^tξ(0)) or ξ(n)= g(Fⁿξ(0)), where F is either a flow or a diffeomorphism with some hyperbolicity and g is a vector function. In this paper we show that both X^ε and a family of diffusions Ξ^ε can be redefined on a common, sufficiently rich probability space so that Esup_0≤t≤T ∣X^ε(t) − Ξ^ε(t)∣^p ≤ Cε^δ, p ≥ 1 for some C, δ > 0 and all ε > 0, where all Ξ^ε, ε > 0 have the same diffusion coefficients but underlying Brownian motions may change with ε.